On the Structure of Involutions and Symmetric Spaces of Quasi Dihedral Group

Abstract

Let $G=Q{D}_{8k}$ be the quasi-dihedral group of order $8n$ and $\theta$ be an automorphism of $Q{D}_{8k}$ of finite order. The fixed-point set $H$ of $\theta$ is defined as $H_{\theta }=G^{\theta }=\{x\in G\mid \theta (x)=x\}$ and generalized symmetric space $Q$ of $\theta$ given by $Q_θ=\{g\in G \mid g=xθ(x)^{-1}~\mbox{for some}~x\in G\}.$ The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,H.Q\ne Q{D}_{8k}.$ the $H$-orbits on $Q$ are obtained under different conditions. Moreover, the formula to find the order of $v$-th root of unity in ${\mathbb{Z}}_{2k}$ for $Q{D}_{8k}$ has been calculated. The criteria to find the number of equivalence classes denoted by $C_{4k}$ of the involution automorphism has also been constructed. Finally, the set of twisted involutions $R=R_{\theta }=\{x\in G\mid \theta (x)=x^{-1}\}$ has been explored.

arXiv commentmajor revision

References in corpus (2)

• On the Structure of Involutions and Symmetric Spaces of Dihedral Groups
• On involutions and generalized symmetric spaces of dicyclic groups